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G = C4⋊(D6⋊C4)  order 192 = 26·3

The semidirect product of C4 and D6⋊C4 acting via D6⋊C4/C22×S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C43(D6⋊C4), D63(C4⋊C4), C121(C22⋊C4), (C2×C12).141D4, (C2×C4).143D12, (C22×S3).9Q8, C2.4(D63D4), C2.4(C12⋊D4), C22.26(S3×Q8), C6.53(C4⋊D4), C2.5(C4.D12), C2.3(D63Q8), (C22×S3).57D4, C22.110(S3×D4), C22.47(C2×D12), (C22×C4).350D6, C6.72(C22⋊Q8), C33(C23.7Q8), C6.C4219C2, C6.37(C42⋊C2), (S3×C23).89C22, C23.305(C22×S3), (C22×C6).351C23, C22.59(D42S3), (C22×C12).143C22, C22.27(Q83S3), (C22×Dic3).58C22, (S3×C2×C4)⋊4C4, (C6×C4⋊C4)⋊5C2, (C2×C4⋊C4)⋊5S3, C6.20(C2×C4⋊C4), C2.21(S3×C4⋊C4), C2.15(C2×D6⋊C4), (S3×C22×C4).1C2, (C2×C6).83(C2×Q8), (C2×C12).85(C2×C4), (C2×C4).154(C4×S3), (C2×D6⋊C4).11C2, (C2×C4⋊Dic3)⋊30C2, (C2×C6).451(C2×D4), C6.42(C2×C22⋊C4), C22.136(S3×C2×C4), C2.13(C4⋊C47S3), C22.66(C2×C3⋊D4), (C2×C6).156(C4○D4), (C2×C4).184(C3⋊D4), (C22×S3).61(C2×C4), (C2×C6).119(C22×C4), (C2×Dic3).96(C2×C4), SmallGroup(192,546)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4⋊(D6⋊C4)
C1C3C6C2×C6C22×C6S3×C23S3×C22×C4 — C4⋊(D6⋊C4)
C3C2×C6 — C4⋊(D6⋊C4)
C1C23C2×C4⋊C4

Generators and relations for C4⋊(D6⋊C4)
 G = < a,b,c,d | a4=b6=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 632 in 234 conjugacy classes, 87 normal (41 characteristic)
C1, C2 [×7], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×7], C22 [×16], S3 [×4], C6 [×7], C2×C4 [×6], C2×C4 [×24], C23, C23 [×10], Dic3 [×4], C12 [×4], C12 [×2], D6 [×4], D6 [×12], C2×C6 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, C4×S3 [×8], C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×6], C2×C12 [×6], C22×S3 [×6], C22×S3 [×4], C22×C6, C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4⋊Dic3 [×2], D6⋊C4 [×4], C3×C4⋊C4 [×2], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C23.7Q8, C6.C42 [×2], C2×C4⋊Dic3, C2×D6⋊C4 [×2], C6×C4⋊C4, S3×C22×C4, C4⋊(D6⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×2], C23, D6 [×3], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C23.7Q8, S3×C4⋊C4, C4⋊C47S3, C12⋊D4, C4.D12, C2×D6⋊C4, D63D4, D63Q8, C4⋊(D6⋊C4)

Smallest permutation representation of C4⋊(D6⋊C4)
On 96 points
Generators in S96
(1 76 16 68)(2 77 17 69)(3 78 18 70)(4 73 13 71)(5 74 14 72)(6 75 15 67)(7 39 91 31)(8 40 92 32)(9 41 93 33)(10 42 94 34)(11 37 95 35)(12 38 96 36)(19 63 27 55)(20 64 28 56)(21 65 29 57)(22 66 30 58)(23 61 25 59)(24 62 26 60)(43 87 51 79)(44 88 52 80)(45 89 53 81)(46 90 54 82)(47 85 49 83)(48 86 50 84)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 95)(8 94)(9 93)(10 92)(11 91)(12 96)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)(43 49)(44 54)(45 53)(46 52)(47 51)(48 50)(55 64)(56 63)(57 62)(58 61)(59 66)(60 65)(67 76)(68 75)(69 74)(70 73)(71 78)(72 77)(79 85)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 44 20 32)(2 45 21 33)(3 46 22 34)(4 47 23 35)(5 48 24 36)(6 43 19 31)(7 67 87 55)(8 68 88 56)(9 69 89 57)(10 70 90 58)(11 71 85 59)(12 72 86 60)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 95 73 83)(62 96 74 84)(63 91 75 79)(64 92 76 80)(65 93 77 81)(66 94 78 82)

G:=sub<Sym(96)| (1,76,16,68)(2,77,17,69)(3,78,18,70)(4,73,13,71)(5,74,14,72)(6,75,15,67)(7,39,91,31)(8,40,92,32)(9,41,93,33)(10,42,94,34)(11,37,95,35)(12,38,96,36)(19,63,27,55)(20,64,28,56)(21,65,29,57)(22,66,30,58)(23,61,25,59)(24,62,26,60)(43,87,51,79)(44,88,52,80)(45,89,53,81)(46,90,54,82)(47,85,49,83)(48,86,50,84), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,95)(8,94)(9,93)(10,92)(11,91)(12,96)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,49)(44,54)(45,53)(46,52)(47,51)(48,50)(55,64)(56,63)(57,62)(58,61)(59,66)(60,65)(67,76)(68,75)(69,74)(70,73)(71,78)(72,77)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82)>;

G:=Group( (1,76,16,68)(2,77,17,69)(3,78,18,70)(4,73,13,71)(5,74,14,72)(6,75,15,67)(7,39,91,31)(8,40,92,32)(9,41,93,33)(10,42,94,34)(11,37,95,35)(12,38,96,36)(19,63,27,55)(20,64,28,56)(21,65,29,57)(22,66,30,58)(23,61,25,59)(24,62,26,60)(43,87,51,79)(44,88,52,80)(45,89,53,81)(46,90,54,82)(47,85,49,83)(48,86,50,84), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,95)(8,94)(9,93)(10,92)(11,91)(12,96)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,49)(44,54)(45,53)(46,52)(47,51)(48,50)(55,64)(56,63)(57,62)(58,61)(59,66)(60,65)(67,76)(68,75)(69,74)(70,73)(71,78)(72,77)(79,85)(80,90)(81,89)(82,88)(83,87)(84,86), (1,44,20,32)(2,45,21,33)(3,46,22,34)(4,47,23,35)(5,48,24,36)(6,43,19,31)(7,67,87,55)(8,68,88,56)(9,69,89,57)(10,70,90,58)(11,71,85,59)(12,72,86,60)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,95,73,83)(62,96,74,84)(63,91,75,79)(64,92,76,80)(65,93,77,81)(66,94,78,82) );

G=PermutationGroup([(1,76,16,68),(2,77,17,69),(3,78,18,70),(4,73,13,71),(5,74,14,72),(6,75,15,67),(7,39,91,31),(8,40,92,32),(9,41,93,33),(10,42,94,34),(11,37,95,35),(12,38,96,36),(19,63,27,55),(20,64,28,56),(21,65,29,57),(22,66,30,58),(23,61,25,59),(24,62,26,60),(43,87,51,79),(44,88,52,80),(45,89,53,81),(46,90,54,82),(47,85,49,83),(48,86,50,84)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,15),(2,14),(3,13),(4,18),(5,17),(6,16),(7,95),(8,94),(9,93),(10,92),(11,91),(12,96),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38),(43,49),(44,54),(45,53),(46,52),(47,51),(48,50),(55,64),(56,63),(57,62),(58,61),(59,66),(60,65),(67,76),(68,75),(69,74),(70,73),(71,78),(72,77),(79,85),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,44,20,32),(2,45,21,33),(3,46,22,34),(4,47,23,35),(5,48,24,36),(6,43,19,31),(7,67,87,55),(8,68,88,56),(9,69,89,57),(10,70,90,58),(11,71,85,59),(12,72,86,60),(13,49,25,37),(14,50,26,38),(15,51,27,39),(16,52,28,40),(17,53,29,41),(18,54,30,42),(61,95,73,83),(62,96,74,84),(63,91,75,79),(64,92,76,80),(65,93,77,81),(66,94,78,82)])

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A···6G12A···12L
order12···22222344444444444444446···612···12
size11···166662222244446666121212122···24···4

48 irreducible representations

dim11111112222222224444
type+++++++++-+++--+
imageC1C2C2C2C2C2C4S3D4D4Q8D6C4○D4C4×S3D12C3⋊D4S3×D4D42S3S3×Q8Q83S3
kernelC4⋊(D6⋊C4)C6.C42C2×C4⋊Dic3C2×D6⋊C4C6×C4⋊C4S3×C22×C4S3×C2×C4C2×C4⋊C4C2×C12C22×S3C22×S3C22×C4C2×C6C2×C4C2×C4C2×C4C22C22C22C22
# reps12121181422344441111

Matrix representation of C4⋊(D6⋊C4) in GL6(𝔽13)

4110000
290000
005000
000800
000010
000001
,
100000
010000
0012000
0001200
0000012
000011
,
1200000
0120000
001000
0001200
0000012
0000120
,
010000
100000
000100
0012000
0000119
000042

G:=sub<GL(6,GF(13))| [4,2,0,0,0,0,11,9,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;

C4⋊(D6⋊C4) in GAP, Magma, Sage, TeX

C_4\rtimes (D_6\rtimes C_4)
% in TeX

G:=Group("C4:(D6:C4)");
// GroupNames label

G:=SmallGroup(192,546);
// by ID

G=gap.SmallGroup(192,546);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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